where \(\Delta ^2=(m_{M}+m_{B})^2\) . We carry out the integral \(\int _{\Delta ^2}^{\infty } ds^\prime \frac{\rho _{H}(s^{\prime },s,u)}{s^\prime -p^{\prime 2}}\)  according to
the hadronic spectral densities in Eq.(REF ) to make the calculations rigorous or robust, rather than just selecting or modeling the hadron representations by hand as in Refs.[1]}, [2]}, [3]}. Now let us write down the quark-hadron duality explicitly,
\(\int _{\Delta _s^2}^{s^0_{M}}ds \int _{\Delta _u^2}^{s^0_{B}}du \frac{\rho _{QCD}(s,u)}{(s-p^2)(u-q^2)}&=& \int _{\Delta _s^2}^{s^0_{M}}ds \int _{\Delta _u^2}^{s^0_{B}}du \int _{\Delta ^2}^{\infty }ds^\prime \frac{\rho _H(s^\prime ,s,u)}{(s^\prime -p^{\prime 2})(s-p^2)(u-q^2)} \nonumber \\&=&\frac{\lambda _{P}\lambda _{M}\lambda _{B}G_{PMB} }{(m_{P}^2-p^{\prime 2})(m_{M}^2-p^2)(m_{B}^2-q^2)} +\frac{C_{P^{\prime }M}+C_{P^{\prime }B}}{(m_{M}^2-p^{2})(m_{B}^2-q^2)} \, , \nonumber \\\) 
